3.453 \(\int x^{-1+n-p (1+q)} \left (a x^n+b x^p\right )^q \, dx\)

Optimal. Leaf size=39 \[ \frac{x^{-p (q+1)} \left (a x^n+b x^p\right )^{q+1}}{a (q+1) (n-p)} \]

[Out]

(a*x^n + b*x^p)^(1 + q)/(a*(n - p)*(1 + q)*x^(p*(1 + q)))

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Rubi [A]  time = 0.0932318, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{x^{-p (q+1)} \left (a x^n+b x^p\right )^{q+1}}{a (q+1) (n-p)} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + n - p*(1 + q))*(a*x^n + b*x^p)^q,x]

[Out]

(a*x^n + b*x^p)^(1 + q)/(a*(n - p)*(1 + q)*x^(p*(1 + q)))

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Rubi in Sympy [A]  time = 10.0923, size = 27, normalized size = 0.69 \[ \frac{x^{- p \left (q + 1\right )} \left (a x^{n} + b x^{p}\right )^{q + 1}}{a \left (n - p\right ) \left (q + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+n-p*(1+q))*(a*x**n+b*x**p)**q,x)

[Out]

x**(-p*(q + 1))*(a*x**n + b*x**p)**(q + 1)/(a*(n - p)*(q + 1))

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Mathematica [B]  time = 0.213892, size = 100, normalized size = 2.56 \[ \frac{x^{-p (q+1)} \left (\frac{a x^{n-p}}{b}+1\right )^{-q} \left (a x^n+b x^p\right )^q \left (b x^p \left (\left (\frac{a x^{n-p}}{b}+1\right )^q-1\right )+a x^n \left (\frac{a x^{n-p}}{b}+1\right )^q\right )}{a (q+1) (n-p)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + n - p*(1 + q))*(a*x^n + b*x^p)^q,x]

[Out]

((a*x^n + b*x^p)^q*(a*x^n*(1 + (a*x^(n - p))/b)^q + b*x^p*(-1 + (1 + (a*x^(n - p
))/b)^q)))/(a*(n - p)*(1 + q)*x^(p*(1 + q))*(1 + (a*x^(n - p))/b)^q)

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Maple [F]  time = 0.351, size = 0, normalized size = 0. \[ \int{x}^{-1+n-p \left ( 1+q \right ) } \left ( a{x}^{n}+b{x}^{p} \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+n-p*(1+q))*(a*x^n+b*x^p)^q,x)

[Out]

int(x^(-1+n-p*(1+q))*(a*x^n+b*x^p)^q,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a x^{n} + b x^{p}\right )}^{q} x^{-p{\left (q + 1\right )} + n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^n + b*x^p)^q*x^(-p*(q + 1) + n - 1),x, algorithm="maxima")

[Out]

integrate((a*x^n + b*x^p)^q*x^(-p*(q + 1) + n - 1), x)

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Fricas [A]  time = 0.267723, size = 103, normalized size = 2.64 \[ \frac{{\left (a x x^{-p q + n - p - 1} x^{n} + b x x^{-p q + n - p - 1} x^{p}\right )}{\left (a x^{n} + b x^{p}\right )}^{q}}{{\left (a n - a p +{\left (a n - a p\right )} q\right )} x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^n + b*x^p)^q*x^(-p*(q + 1) + n - 1),x, algorithm="fricas")

[Out]

(a*x*x^(-p*q + n - p - 1)*x^n + b*x*x^(-p*q + n - p - 1)*x^p)*(a*x^n + b*x^p)^q/
((a*n - a*p + (a*n - a*p)*q)*x^n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+n-p*(1+q))*(a*x**n+b*x**p)**q,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a x^{n} + b x^{p}\right )}^{q} x^{-p{\left (q + 1\right )} + n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^n + b*x^p)^q*x^(-p*(q + 1) + n - 1),x, algorithm="giac")

[Out]

integrate((a*x^n + b*x^p)^q*x^(-p*(q + 1) + n - 1), x)